Bose-Einstein Condensation Nuclear Fusion:

Theoretical Predictions and Experimental Tests

Yeong E. Kim

Purdue Nuclear and Many-Body Theory Group

Department of Physics, Purdue University

West Lafayette, IN 47907

Presented at

The 15th International Conference on Condensed Matter Nuclear

Science (ICCF-15), Rome, Italy, October 5 – 9, 2009

References: Y. E. Kim, Naturwissenschaften (2009) 96: 803-811 (published

online14 May 2009) and references therein

1

Experimental Observations

• (D+D) fusion in free space (E ≥ 10 keV):

{1} D + D→ p + T + 4.03 MeV

{2} D + D→ n + 3He + 3.27 MeV

{3} D + D→ 4He + γ + 23.8 MeV

R{1}≈R{2} and R{3}/R{1}≈10-6

● (D+D) fusion in metal (E ≤ 0.1 eV) (m represents a host metal

lattice or metal particle) :

{4} D(m) + D(m) → p(m) + T(m) + 4.03 MeV(m)

{5} D(m) + D(m) → n(m) + 3He(m) + 3.27 MeV(m)

{6} D(m) + D(m) → 4He(m) + 23.8 MeV (m)

Fusion rate R{6} for {6} is much greater than rates R{4} and R{5}22

(D+D) fusion in metal (E< 0.1 eV) (m represents a host metal lattice or metal

particle) :

{4} D(m) + D(m) → p(m) + T(m) + 4.03 MeV(m)

{5} D(m) + D(m) → n(m) + 3He(m) + 3.27 MeV(m)

{6} D(m) + D(m) → 4He(m) + 23.8 MeV

Experimental Observations (as of 2008) (not complete)

From both electrolysis and gas loading experiments

[1] The Coulomb barrier between two deuterons are suppressed

[2] Excess heat production (the amount of exess heat indicates its nuclear origin)

[3] 4He production comensurate with excess heat production, no 23.8 MeV gamma ray

[4] Production of hot spots and micro-scale crators on metal surface

[5] Detection of radiations

[6] Production of nuclear ashes with anomalous rates: R{4} << R {6} and R {5} << R{6}

[7] “Heat-after-death”

[8] Requirement of deuteron mobility (D/Pd > 0.9, electric current, pressure gradient,

etc.)

[9] Requirement of deuterium purity (H/D << 1)

[10] More tritium is produced than neutron R(T) >> R(n)3

Based on a single physical concept, can we come

up with a consistent physical theory which could

explain all of the ten experimental observations ?

Deuterons become mobile in metal when electric

current (Coehn 1929), and/or pressure gradient is

applied !

Explore a concept of “nuclear” Bose-Einstein

Condensation of deuterons in metal for

developing a consistent physical theory to explain

the experimental observations, [1] through [10].4

Requirement for

Bose-Einstein

Condensation

(BEC):

λDB > d

where d is the

average distance

between

neighboring

two Bosons

55

6

Created in 1995 by C. Wieman,

E. Cornell, W. Ketterle, et al.

Nobel Prize in 2000

Atomic BEC vs. Nuclear BEC

BEC Requirement: λDB > d, λDB = or υkT < υc

Atomic BEC: d ≈ 7 x 103 Å = 0.7 μm (for nRb = 2.6 x 1012/cm3)

υc ≈ 0.6 cm/sec (υkT≈ 0.58 cm/sec, at T ≈ 170 n Kelvin)

( ~ 2000 atoms in BEC out of ~ 2 x 104 atoms 10 % in BEC)

(1) Increase λDB by slowing down neutral atoms

using laser cooling and evaporation cooling

Nuclear BEC: d ≈ 2.5 Å (for nD = 6.8 x 1022/cm3 in metal)

υc ≈ 0.78 x 105 cm/sec (υkT ≈ 1.6 x 105 cm/sec at T= 300 Kelvin)

(1) Increase λDB by slowing down charged deuterons using

electromagnetic fields, pressure gradient, and/or cooling

(2) Decrease d by compression using ultrahigh pressure

device such as Diamond Anvil Cell (DAC)

h

mυ

7

Fraction F of Deuterons in the BEC State in Metal at

Various Temperatures

At 300ºK with Ec= 0.00655 eV corresponding to

F (Ec) = ~0.084 (8.4% !), ( ~10% for the atomic BEC case)

Since mobile deuterons in metal are localized within several metal

lattice sites, 8.4 % of mobile deuterons with υ ≤ υc

(satisfying λDB > d) may not encounter each other frequently

enough to form the BEC.

Need to increase 0.084 (8.4%) to 0.28 (2/7 or 28%) (which is

based on a geometrical argument), or

Collect 8.4% into localized regions by applied EM fields.

At 77.3ºK (liquid nitrogen), F(Ec) = ~0.44 (44%)

using Bose-Einstein distribution

At 20.3ºK (liquid hydrogen) F(Ec) = ~0.94 (94%)

using Maxwell-Boltzmann distribution.

2.5 ,dB

d

8

Boson-Einstein Condensation (BEC) Mechanism

N-Body Schroedinger Equation for the BEC State

For simplicity, we assume an isotropic harmonic

potential for the deuteron trap.

N-body Schroedinger equation for the system is

2 2N N2 2

iii ji=1 i=1 i<j

1 eH= Δ + mω +r

2m 2 r -r

where m is the rest mass of the nucleus.

In presence of electrons, we use the shielded Coulomb potential

(Debye screening)

(1)

(2)

H E

where Hamiltonian is given by

BEC Mechanism

9

The total fusion rate Rt is given by

(19)

1/ 2

2D

t trap trap trap D

N 1 3R N R R Vn

N 4

Only one unknown parameter is the probability of the BEC

ground-state occupation, .

→ Observation [1] The Coulomb barrier between two

deuterons are suppressed.

Total Reaction Rate

10

3/ 2 1/ 22

trap D3

trap

1 3 N 1 3R n N

2 4D

where A = 2Sħ / (π me2) with S = 55 keV-barn, Dtrap is the average

diameter of the trap, Dtrap= 2 <r>, ND is the total number of deuterons, N

is the number of deuterons in a trap, and nD is the deuteron density.

For a single trap (or metal particle) containing N deuterons, we

have for primary reactions: leading to secondary reactions

●

●

●

4N 2 D's D D ψ* He N 2 D's Q 23.84 MeVBECψ

leading to micro-scale explosions or “melt-down”

where ψBEC is the Bose-Einstein condensate ground-state (a coherent

quantum state) with N deuterons, and ψ* are continuum final states.

Excess energy (Q value) is absorbed by the BEC state and shared

by reaction products in the final state.

Observation [2] Excess heat production and [3] 4He

production, without 23.8MeV gamma rays.

3D fusion (D + D + D) and 4 D fusion are possible, but their

fusion rates are expected to be much smaller than that of the 2D

fusion, leading to secondary effects. 11

4D's ψ* ( /2) He Q ( /2) 23.84 MeVBECψ N N N

Conversion of nearly all deuterons to 4He by BECNF in

metal grains and particles in the host metal

Sustained BECNF and heat production

Episodes of “Melt Down” reported by

Fleischmann and others

• Excess energies (Q) leading to a micro/nano-

scale explosion creating a crater/cavity and a

hot spot with firework-like tracks.

• Size of a crater/cavity will depend on number of

(D + D) fusions occuring simultaneously in BEC

states.

• Observation [4] Production of hot spots and

micro-craters. 12

4D's ψ* ( / 2) He Q ( / 2) 23.84 MeVBECψ N N N

leading to secondary reactions

Total Momentum Conservation

• Initial Total Momentum:

• Final Total Momentum:

• <T> is the average kinetic energy.

• ~1 keV (up to 23.8 MeV) deuterons from {6} lose

energies by electrons and induce X-rays, γ-rays, and

Bremsstrahlung X-rays.

Observation [5] Detection of radiations.

N-2 4 4DD He He

Q{6}{6} P 0, T T

N

ND

P 0

1313

4N 2 D's D D ψ* He N 2 D's Q 23.84 MeVBECψ

Selection Rule for Two-Species Case

(m is mass number approximately given in units of the nucleon mass) 2

2

1

1

m

Z

m

Z

3

1 2 2 2 2

3

1 2 2 2 2

Z (D) Z (p) Z (T) Z (n) Z ( He)1 1 2, 1, , 0,

m (D) 2 m (p) m (T) 3 m (n) 3m ( He)

•Reactions {4} and {5} are forbidden/suppressed→ reaction rates are small

{4} D(m) + D(m) → p(m) + T(m) + 4.03 MeV (m)

{5} D(m)+D(m) → n(m) + 3He(m) + 3.27 MeV (m)

____________________________________________________

•Reaction {6} is allowed → reaction rate is large

{6} D(m)+D(m)→4He(m) + 23.8 MeV (m)

→ This explains Observation [6] R(4) << R(6) and R(5) << R(6).

4

1 2

4

1 2

Z Z ( He)(D) 1= =

m (D) 2m ( He)

Selection Rule

14

• Heat after Death (Observation [7])

Because of mobility of deuterons in Pd nanoparticle traps, a

system of ~1022 deuterons contained in ~ 1018 Pd nanoparticle

traps is a dynamical system (in 3 g of 5 nm Pd nanoparticles).

BEC states are continuously attained in a small fraction of the

~1018 Pd particle traps and undergo BEC fusion processes, until

the formation of the BEC state ceases.

• Deuteron Mobility Requirement (Observation [8])

D/Pd ≥ ~0.9 is required for sustaining deuteron mobility in Pd.

Electric current or pressure gradient is required.

• Deuterium Purity Requirement (Observation [9])

Because of violation of the two-species selection rule,

presence of hydrogens in deuteriums will surpress the formation

of the BEC states, thus diminishing the fusion rate due to the BEC

mechanism.15

(c) Mixed oxides of PdZr

0 500 1000 1500

0

0.4

0.8

1.2

0

0.4

0.8

1.2

Time [min]

Ou

tpu

t p

ow

er [

W]

Press

ure [

MP

a]

Power (D2) Power (H2) Pressure (D 2) Pressure (H 2)

Fig. 3(c): A. Kitamura et al., Physics Letters A, 373 (2009) 3109-3112.

16

•Consistent with [8] the requirement of deuteron mobility

(D/Pd > 0.9, electric current, pressure gradient, etc.)

•Output power of 0.15 W corresponds to Rt ≈ 1 x 109 DD fusions/sec

for D+D → 4He + 23.8 MeV

BEC Mechanism on Reactions {4} and {5}

(Secondary Reactions)

• R{4} << R{6}, R{5} << R{6}, due the selection rule

where neutron, n(m), is at energies ~keV.

• ~keV neutron(m) from Reaction {5} can undergo further reactions,

{12}, and/or {13} below:

{12} n(m) + D(m) (in BEC State) → T(m) +6.26 MeV(m)

{13} n(m) + D → T + γ + 6.26 MeV

→ Reactions {12} and {13} produce more tritiums than neutrons, R(T) >

R(n).

→ R(T) > R(3He)

This explains Observation [10] more tritium is produced

than neutron.

)(8.23)(}6{

)(27.3)(}5{

)(03.4)(}4{

4

3

mMeVmHemDmD

mMeVmTmnmDmD

mMeVmHempmDmD

Selection Rule

1717

Experimental Tests of Predictions of BECNF theory

1. Tests based on the average size of metal particles

2. Tests for reaction-rate increases by applied EM fields

3. Tests for resistivity change

4. Tests for scalability

Basic Fundamental Tests of “Nuclear” Bose-Einstein

Condensation of Deuterons in Metal

5. Ultrahigh pressure experimental tests

6. Low temperature experimental tests

These experimental tests are needed

(1) to improve and/or refine the theory, and also

(2) to achieve 100 % reproducibility for experimental results, and

for possible practical applications.

18

Experimental Tests 1. Tests based on the average size of metal nanoparticles

● The total fusion rate is given by

(19)

where Ntrap is the total number of traps, ND is the total number of deuterons, N

is the number of deuterons in a trap, and Ω is the probability of the BEC

state occupation.

For the case of Ω proportional to the ratio of surface area/ volume of each

particle:

Rt (smaller Pd particles) > Rt (larger Pd particles)

● The above theoretical prediction [Kim, Naturwissenschaften 96 (2009)

803-811 (14 May 2009)] is experimentally confirmed by A. Kitamura et

al./ Physics Letters A 373 (2009) 3109-3112 (4 July 2009)

1/ 2

2D

t trap trap trap D

N 1 3R N R R Vn

N 4 π

t t

t t

R 5nm R 2nm2, 5,

R 10nm R 10nmetc. t trap

trap

1R (D )

D

1919

1/ 3 1

trapN or D

(a) 0.1-mmf Pd

0 500 1000

0

0.4

0.8

1.2

0

0.4

0.8

1.2

Time [min]

Ou

tpu

t P

ower

[W

]

Pre

ssu

re [

MP

a]

Power (D2) Power (H2) Pressure (D2) Pressure (H2)

(c) Mixed oxides of PdZr

0 500 1000 1500

0

0.4

0.8

1.2

0

0.4

0.8

1.2

Time [min]

Ou

tpu

t p

ow

er [

W]

Pre

ssu

re [

MP

a]

Power (D2) Power (H2) Pressure (D 2) Pressure (H 2)

A. Kitamura et al./

Physics Letters A 373

(2009) 3109-3112

20

Rt (10.7-nmφPd) > Rt (0.1μmφPd)

5 g of

100 nm Pd particles

3 g of

10.7 nm Pd particles

t

t

R 10.7 nm Pd

R 100 nm PD

100 nm9.3

10.7 nm

2. Tests for reaction-rate increases by applied EM fields

Increase of reaction-rate is expected by increase of BEC deuteron

fraction which can be accomplished by applied EM fileds

(electric currents (AC or DC), external electric field, external

magnetic field, etc.)

3. Tests for resistivity change

Measure resistivity change which is expected when BEC occurs.

4. Tests for scalability

Examples:

etc. 10particles Pd 3g

particles Pd 30g

same for the

,

t

t

traptraptraptrapt

R

R

RNRNR

keVN

Q~

}5{

2121

5. Proposed Basic Fundamental Tests of “Nuclear” BEC - I

Schematics of the core of a diamond anvil cell.

The diamond size is a few millimeters at most

Ultrahigh pressure experimental tests

22

● Apply electric current

through the sample.

● Sudden change in the

resistivity is expected

when deuterons form a

BEC state at some

pressure.

● Emission of radiations

and neutrons may be

expected when BECNF

occurs.

● Possibility of using

laser beam to measure

Raman scattering

frequency shifts.

(Deuterated Pd, 50-300 μm)

22

6. Proposed Basic Fundamental Tests of “Nuclear” BEC - II

Low temperature experiments

● Test of theoretical prediction Rt(Tlow) > Rt(Thigh)

● At liquid nitrogen temperature (77.3 Kelvin), the fraction F(Ec)

of mobile deuterons in metal satisfying λDB > d ≈ 2.5 Å

or E < Ec = 0.00655 eV is F(Ec) = ~ 0.44 (~44% !)

● At liquid hydrogen temperature (20.3 Kelvin), F(Ec) = ~ 0.94

(~94 % !), using Maxwell-Boltzmann distribution.

● Apply electric current through the sample.

● Change in the resistivity is expected when deuterons form a

BEC state at some lower temperatures

● Emission of radiations and neutrons may be expected when

BECNF occurs, as secondary effects.

● Shifts in Raman scattering frequencies are expected when

BEC occurs232323

Other Potential Applications of the Concept of

Bose-Einstein Condensation of Deuterons in Metal

1. Transmutation

2. Transient Acoustic Cavitation Fusion

3. High Temperature Superconductivity of

metal/alloy hydrides/deuterides

24

Conclusions and Summary● BECNF Theory provides a consistent conventional theoretical

description of the experimental observations, [1] through [10].

● Experimental tests of a set of six (6) key theoretical predictions

are proposed including two basic fundamental experimental tests

of the concept of “nuclear” Bose-Einstein condensation of

deuterons in metal

● Experimental tests of the predictions of the BECNF theory are

needed in order (1) to improve and/or refine the theory, and also

(2) to achieve 100 % reproducibility for practical applications.

● If the theoretical predictions are all confirmed experimentally, the

concept of Bose-Einstein condensation of deuterons in metal may

become a new discovery. 25

Backup Slides

26

Example with 3g of 50 Pd particles

• Total number of Pd atoms in 3g, NPd = 1.7 x 1022 Pd atoms

NPd = 3g × (6.02 × 1023)/106.4g ≈ 1.7 × 1022 Pd atoms

For ND ≈ NPd, ND = 1.7 × 1022 D atoms

• The number density of Pd,

nPd = 12.03 g cm-3 × (6.02 × 1023)/106..4 g ≈6.8 × 1022cm-3

• One Pd particle of diameter ~ contains

deuterons

• In 3g of Pd particles, the total number of Pd particle traps is

particle traps

• For comparison, ~ 2000 atoms are trapped for the atomic case.

o

A

DPd ncmn 3221086.

o

A50 4450A506

3o

DnN

18

3

22

108310454

1071

.

.

.

N

N

N

N PdD

27

Theoretical derivations of BEC nuclear fusion rates are given in the

following references:

Approximate Solution of Many-Body Schroedinger Equation1. Y.E. Kim and A.L. Zubarev, “Equivalent Linear Two-Body Method for Many-

Body Problems", J. Phys. B: At. Mol. Opt. Phys. 33, 55 (2000).

2. Y.E. Kim and A.L. Zubarev, “Equivalent Linear Two-Body Method for Bose-Einstein Condensates in Time-Dependent Harmonic Traps", Physical Review A66, 05362 (2002).

Optical Theorem Formulation of Nuclear Reactions3. Y.E. Kim, Y.J. Kim, A.L. Zubarev, and J.-H. Yoon, “Optical Theorem

Formulation of Low-Energy Nuclear Reactions", Physical Review C55, 801 (1997).

Nuclear Fusion Rates for Deuterons in the BEC State4. Y.E. Kim and A.L. Zubarev, “Nuclear Fusion for Bose Nuclei Confined in Ion

Traps", Fusion Technology 37, 151 (2000).

5. Y.E. Kim and A.L. Zubarev, “Ultra Low-Energy Nuclear Fusion of Bose Nuclei in Nano-Scale Ion Traps", Italian Physical Society Conference Proceedings, Vol. 70, (May 2000), pp. 375-384.

28

Fusion Reaction Rates

Our final theoretical formula for the nuclear fusion rate

Rtrap for a single trap containing N deuterons is given by

(18)

BEC mechanism

2

3

3ω α

4π

c N

m r

2 2B B

where r is the radius of trap/atomic cluster, r = Ψ r Ψ ,

B is given by B=3Am/ 8πα c ,

N is the average number of Bose nuclei in a trap/cluster.

A is given by A=2Sr / π , where r = / 2μe , μ=m/2,

S is th3

e S-factor for the nuclear fusion reaction between two deuterons (for D(D,p)T and D(d,n) He reactions, S 55 keV-barn)

2ωtrap

R BN

29

All constants are known except Ω, which is the probability of the

BEC ground state occupation

Equivalent Linear Two-Body (ELTB) Method(Kim and Zubarev, Physical Review A 66, 053602 (2002))

For the ground-state wave function , we use the following

approximation (3 1) / 2

( )( )( ,... )N Nr r

where1/ 2

2

1

N

i

i

r

It has been shown that approximation (3) yields good results for the case of

large N (Kim and Zubarev, J. Phys. B: At. Mol. Opt. Phys. 33, 55 (2000))

By requiring that must satisfy a variational principle with a

subsidiary condition , we obtain the following Schrödinger

equation for the ground state wave function ()

* 0H d * 1d

2 2 22 2

2 2

(3 1)(3 3)[ ( )]

2 2 2 4

d m N NV E

m d m

(3)

(4)

2 (3 / 2)( )

3 2 (3 / 2 3/ 2)

N NV

N

where (5)

3030

Optical Theorem Formulation of Nuclear Fusion Reactions (Kim, et al. Physical Review C 55, 801 (1997))

In order to parameterize the short-range nuclear force, we use the optical theorem

formulation of nuclear fusion reactions. The total elastic nucleus-nucleus amplitude

can be written as( ) ( ) ( )cf f f

where is the Coulomb amplitude, and can be expanded in partial waves( )cf ( )f

2 ( )( ) (2 1) (cos )cli n el

l

f l e f P

In Eq. (7), is the Coulomb phase shift, , and

is the l-th partial wave S-matrix for the nuclear part.

c

( 1) / 2n(el) nf S ik ns

For low energy, we can write (optical theorem)

( )Im4

n el rkf

where is the partial wave reaction cross section.r

In terms of the partial wave t-matrix, the elastic scattering amplitude, can be

written as

( )n elf

( )

2 2

2n el c cf tk

m

where is the Coulomb wave function. c

(6)

(7)

(8)

(9)

3131

Parameterization of the Short-Range Nuclear Force

For the dominant contribution of only s-wave, we have

( )

0Im4

n el rkf

( )

0 0 0 02 2

2n el c cf tk

m

Where is conventionally parameterized as r

2r Se

E

2

2

1, , / 2

2 2B

B

r mkr e

mm

, is the “Gamow” factor,

2e

From the above relations, Eqs. (10), (11), and (12), we have

0 0 02 2

2Im

4

r c ckt

k

m

For the case of N Bose nuclei, to account for a short range nuclear force between two

nuclei, we introduce the following Fermi pseudo-potential ( )FV r

0Im Im ( ) ( )2

F At V r r

where the short-range nuclear-force constant A is determined from Eqs. (12) and (13) to

be . 2 /BA Sr

For deuteron-deuteron (DD) fusion via reactions D(d,p)T and D(d,n)3He,

the S-factor is S = 110 KeV-barn.

and

(10)

(11)

(12)

(13)

(14)

and S is the S- factor for the nuclear fusion reaction between two nuclei.

3232

Derivation of Fusion Probability and Rates

For N identical Bose nuclei confined in an ion trap, the nucleus-nucleus fusion rate is

determined from the trapped ground state wave function as Im2 i j ij

t

tR

where is given by the Fermi potential Eq. (14), . Im ijt

From Eq. (15), we obtain for a single trap

3

4t B

cR B Nn

m

where is the probability of the ground state occupation, 2 3/ , /Be c n N r

is Bose nuclei density in a trap, and with

For the case of multiple ion traps (atomic clusters or bubbles), the total ion-trap nuclear

fusion rate R per unit time and per unit volume, can be written as

3

4t B

cR n B Nn

m

where nt is a trap number density (number of traps per unit volume) and N is the

average number of Bose nuclei in a trap.

Im ( ) / 2ijt A r

(15)

(16)

(17)

3 /8B Am c 2 /BA Sr

3333

Selection Rule for Two Species Case

Mean-Field Theory of A Quantum Many-Particle System

(Hartree-Fock Theory)

We consider a mixture of two different species of positively charged

bosons, with N1 and N2 particles, charges Z1≥0 and Z2≥0, and rest

masses m1 and m2, respectively. We assume

The mean-field energy functional for the two-component system is

given by generalization of the one-component case

(20)

where

is density of specie i, and (21)

,int

2

1

EEEi

i

,2

22

i

i

im

rdE

yx

ynZynZxnZxnZydxd

eE

22112211

2

int2

,2

iin .ii Nrnrd

Two Species Case

34

2 2

i i iV (r) m ω r / 2.

The minimization of the energy functional, Eq. (20), with subsidiary

conditions, Eq. (21), leads to the following time-independent mean-field

equations.

(22)

where

(23)

and μi are the chemical potentials, (general thermodynamics

identity).

In the Thomas-Fermi (TF) approximation (neglects the kinetic energy terms

in Eq. (22)), Eq. (22) reduce to

(24)

which leads to the selection rule (derivation in a backup slide),

,2

22

rrWVrm

iiiiii

i

m

,/2121

222 ynyrynynZZynZyderW iiii

.i

iN

E

m

Two Species Case

iii WV m

35

1 2

1 2

Z Z

m m

Selection RuleFor the BEC mechanism for LENR, we obtain nuclear charge-mass selection rule (approximate).

Nuclear mass-charge selection rule:

We can obtain from Eq. (24) that

Since μi are independent of r, we have proved that Eq. (23) has non-trivial solution if and only if

(25)

If we assume ω1=ω2,

and we have from Eq. (25), λ=m2Z1/m1Z2=1 or

(26)

2

2

1

1

m

Z

m

Z

Selection Rule

iii WV m

.2

2

11

1

2

2

11

2

221

1

22

2r

m

Z

Z

m

m

Z

Z

mm

.12

2

11

1

2

22 Zm

Zm

,2

3 (G.S.)

2

3(G.S.) 2

21

1

ωE

ωE

or01

2

2

11

2

22 ,

Z

Z

m

m

3636

Fraction of Deuterons in the BEC State in Metal at

Room TemperatureFor Bose-Einstein distribution, the distribution function is given by

where with the deuteron density, nD.

Using the density of (quantum) states N(E) given by

the total number of N can be calculated as

A fraction F(Ec) of N deuterons below the critical energy Ec satisfying

can be calculated as

For Ec= 0.00655 eV corresponding to ,

F (Ec) = 0.084 (8.4% !), compared to ~10% for the atomic

BEC case. 37

/

1( )

1BE E kT

n Ee e

3/ 2 32 /

De mkT n h

3

4( ) 1/23V

N E dE E dEh

(2m )

/0 0

( )( ) ( )

1BE E kT

N E dEn E N E dE

e eN

/dB dB

d h m

2.5dB

d

0

1( ) ( ) ( )

cE

c BEF E n E N E dE

N

37